Minimax Nonparametric Two-Sample Test under Adversarial Losses

In this paper, we consider the problem of two-sample hypothesis testing that aims at detecting the difference between two probability densities based on finite samples. The proposed test statistic is constructed by first truncating a sample version of a negative Besov norm and then normalizing it. Here, the negative Besov norm is the norm associated with a Besov space with negative exponent, and is shown to be closely related to a class of commonly used adversarial losses (or integral probability metrics) with smooth discriminators. Theoretically, we characterize the optimal detection boundary of two-sample testing in terms of the dimensionalities and smoothness levels of the underlying densities and the discriminator class defining the adversarial loss. We also show that the proposed approach can simultaneously attain the optimal detection boundary under many common adversarial losses, including those induced by the ℓ₁, ℓ₂ distances and Wasserstein distances. Our numerical experiments show that the proposed test procedure tends to exhibit higher power and robustness in difference detection than existing state-of-the-art competitors.